Secondary Mathematical Challenges

Secondary Mathematical Challenges

Welcome to the 2021-2022 Scottish Secondary Mathematical Challenges. This package contains

This Welcome Page (including Section Information)The Secondary GuidelinesRound 1 QuestionsInformation About PaymentsHow to Enter your School and Pupil Names on the Marks WebsiteThe Annual PosterA Book Order Form

Once again, the name of the Section Organiser is not on the question paper. Their details are on thewebsite but are repeated here for convenience:

Section 1Aberdeen City; Aberdeenshire; Highland; Moray; Orkney Islands; Shetland Islands; Western IslesDr William Turner ([email protected])Mathematical ChallengeDepartment of Mathematical Sciences, University of Aberdeen,Aberdeen AB24 3UE

Section 2Angus; Clackmannanshire; Dundee City; Falkirk; Fife; Perth & Kinross; StirlingDr Jean Reinaud ([email protected])Mathematical Institute, University of St Andrews, St Andrews, Fife KY16 9SS

Section 3East Lothian; Edinburgh City; Midlothian; Scottish Borders; West LothianThe contact details for 2021-22 have yet to be finalised.If you have any questions, please contact Bill Richardson ([email protected]).

Section 4Argyll & Bute; Dumfries & Galloway; East Ayrshire; East Dunbartonshire; East Renfrewshire; Glasgow City; Inverclyde; North Ayrshire; North Lanarkshire; Renfrewshire; South Ayrshire; South Lanarkshire; West DunbartonshireScottish Mathematical Challenge Organiser ([email protected]),Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH

The competition timetable for 2021-2022 is as follows:

Set No.Last date for receipt of questions by schools

Last date for receiptof solutions from pupils

I Friday 20 August 2021 Friday 5 November 2021II Friday 26 November 2021 Friday 11 February 2022

If there are organisational difficulties you may contact me, Bill Richardson, ([email protected]).

The Scottish Mathematical CouncilMC homepage: www.scot-maths.co.uk/

MATHEMATICAL CHALLENGE 2021−2022A national problem solving competition for schools in Scotland

S E C O N D A R Y D I V I S I O N SGUIDELINES FOR TEACHERS

1. Mathematical Challenge is aproblem-solvingcompetitionwhich goesbackto 1976.TheChallengeis opento all studentseducatedin Scotland.Its aim is to promotemathematicsasa sourceof interestand pleasurableachievementthrough challenging problems which require only elementarytechniques and simple logic.Please ensure that all teachers involved in the competition see these Guidelines.

How Mathematical Challenge operates 2. Therearefour divisions:JUNIOR for Sl andS2,MIDDLE for S3 andS4,SENIORfor S5 andS6,

and PRIMARY (for which a separate circular is available).Pupils may enter only one division and must specify that division on their first entry.Pleasecontactyour local organiser,whosenameandaddressareon theproformaon page2 of eachproblemsheetandin theContactssectionof theWebpages,if thereis anydoubtaboutdivisions,orif further information is required.

3. There are no written examinations.For the Junior, Middle and Senior Divisions, two sets of fiveproblemseachwill be availablefor schoolsto downloadaccordingto a timetableoutlinedin §13 below.Problemsfor differentdivisionswill be on separatesheets.Someproblemsmaybecommonto differentdivisions.Theproblemswill alsobeavailablefrom theMathematicalChallengeWeb pages(seeabovefor address).

4. A registration fee is required from participating schools.For a secondaryschoolthe feeis £20for the first 10 entrants and half this amount for eachsubsequentbatch of 10 entrants or partthereof. A fee form is included with the downloadable pack of materials.For individual participants NOT entering through a school, the fee is £8.

Entries and Marking5. Entries must be the unaided efforts of individual pupils. Group working is not appropriate in

Mathematical Challenge.Participantsmayconsultbooksor the internetfor informationon factsoron how to tackle problems.Whilst teachersor parentsmay give guidanceon interpretationofwording,they shouldnot be involved in the solution of a problem. Furthermore,the work shouldnot interfere with normal teaching and in no circumstances should it be a class assignment.

6. All Sectionsmustusethe softwarepackageto assistin theprocessingof the results. A Record ofEntries must be made electronically by the school, or it will not be possible to process the results.

• Go to the marks website: https://www.scottishmathschallenge.org.uk/• Choose “School Login” and enter your login details or “Register here” to set up a new account.• Whenyou haveloggedin, go to "Add/Edit Entrants"_ enterthenamesandschoolyearof

each entrant.• The marks will eventually appear on the “Marks page”.• Messages from the organiser may also appear there on the first page from time to time.

Useapapercopyof the‘Printableversionof detailsandentrants’from themarkswebsiteasa coversheetfor theschool'sentries. This containsthe schooldetailsandthealphabeticallist of entrantsineachsection,asenteredon thewebsite. All entriessubmittedwill bemarkedevenif earlierproblemsets are missed.

7. Entries will not be returned. Entrants should keep a copy of their solutions. The ScottishMathematical Council reserves the right to publish good solutions in its Journal.

8. Participants should explain their solutions as fully as they can. Marks will be given forexplanationsof answersnot just for the answersthemselves.We should be most grateful ifteachers would stress this point. Incomplete or incorrect answers may gain some credit.In outline, the marking scheme for each problem is as follows:

4 : a completely correct solution, with full explanation.3 : a solution, with explanation,which is correct apart from a minor slip or

omission of a special case.2 : a solution with explanationwhich containsa seriouserror or omission,but

which nevertheless involves good ideas.1 : thereis an indicationof an interestingideaor method,but notnecessarilyone

which could lead to a correct solution.A bonus mark maybe givenfor a completelycorrectsolution,with full explanation,which containsadditional good ideas, such as a successful generalisation of the problem.A solution in which an answer is given without any explanation will normally be awarded nomarks, evenif the answer is correct. However,correctworking may be acceptedasproviding anexplanation, so long as the various steps are clear.

9. No problems set in Mathematical Challenge require the use of a computer package(e.g. aspreadsheet)to obtaina correctsolution.If computersoftwareis used,thena propermathematicalexplanation of its use is essential.

Awards 10.Therearethreeclassesof award:Gold, Silver andBronze. Awardwinnerswill be selectedprimarily

onthebasisof thetotal numberof marksobtainedoverbothsetsof Problems.Specialcircumstancesfor individual entrants may be taken into account.

11.All awardwinnerswill qualify for certificates.Whereanawardceremonycanbearranged,themostsuccessfulentrantswill be invited to attendto receivetheir certificatesandMathematicalChallengemugs. Certificates not presented at a ceremony will be sent by post.

Important notes 12. Large numbersof entriescan imposea considerablestrain on markersand on organisers.Local

organisersmayhaveto setlimits on thetotalnumbersof entriesperschool.Schoolssubmittinglargenumbersof entriesmaybe askedto provideadditionalmarkers.Any suchmarkerswould not markentries from their own schools.

13. The timetable for 2021-2022 is as follows:

Set No. Last date for receiptof questions by schools

Last date for receiptof solutions from pupils

I Friday 20 August 2021 Friday 5 November 2021II Friday 26 November 2021 Friday 11 February 2022

14.The problems of earlier sessionsform an excellentresource.Thosefor theyears1991-92to 2005-2006, including solutions,are availablein the books Mathematical Challenges III, MathematicalChallenges IV, Mathematical Challenges V andMathematical Challenges VI whicharepublishedbyThe Scottish Mathematical Council. Copies can be obtained from Bill Richardson,Kintail,Longmorn, Elgin IV30 8RJ, prices £7.50, £8, £8, £8 respectively.

In addition, it seemsunlikely that any further bookswill be printedso questionsandsolutionsfor2006-2018 can be accessed at: www.wpr3.co.uk/MC-archive/Comments on the usefulness of these to [email protected] would be welcome.

15.For otherinformation,pleasecontactyour local organiser,whosenameandaddressaregiven in theContacts section of the Mathematical Challenge Web pages

www.scot-maths.co.ukas well as on the materials download menu page

www.wpr3.co.uk/MC/materials

The Scottish Mathematical Councilwww.scot-maths.co.uk

MATHEMATICAL CHALLENGE 2021−2022Entries must be the unaided efforts of individual pupils.

Solutions must include explanations and answers without explanation will be given no credit.Do not feel that you must hand in answers to all the questions.

CURRENT AND RECENT SPONSORS OF MATHEMATICAL CHALLENGE AREThe Edinburgh Mathematical Society, The Maxwell Foundation,

The London Mathematical Society and The Scottish International Education Trust.The Scottish Mathematical Council is indebted to the above for their generous support and gratefully acknowledges financial and

other assistance from schools, universities and education authorities.Particular thanks are due to the Universities of Aberdeen, Edinburgh, Glasgow, St Andrews, Stirling, Strathclyde

and to George Heriot's School, Gryffe High School and Kelvinside Academy.

Junior Division: Problems 1

J1. In the diagram (not to scale) the rectangle is divided into nine smaller rectangles. The areas of fiveof the smaller rectangles are given. Determine the area of the rectangle labelled .R

5

3

4

2

2

R

J2. Forest Rovers, Southside United, Hilltown Thistle and Valley Wanderers were to play each other atfootball. After some of the matches had been played, a table showing some details of matchesplayed, won, lost, drawn etc looked like this:

Played Won Lost Drawn Goals for

Goals against

Points

Forest Rovers 2 1 0

Southside United 0

Hilltown Thistle 1

Valley Wanderers 0 0 4 2 6

3 points are given for a win and 1 for a draw. Complete the table and find the score in each matchplayed, explaining how you worked it out.

J3. and are two towns connected by a single road which crosses a bridge over a wide river. Jamesleaves at 10.38 am and walks along the road to at uniform speed, reaching at 1.50 pm. On thesame day Isla leaves at 9.20 am and walks along the road to at uniform speed, reaching at 12noon. They arrive at their nearest end of the bridge at the same time. James leaves the bridge oneminute later than Isla.

A BA B B

B A A

At what time did they reach the bridge?

SEE OVER FOR QUESTIONS J4 & J5.

SMC

Mathematical Challenge Problems 1JUNIOR DIVISION 2021-2022

PLEASE USE CAPITALS TO COMPLETE

SURNAME

OTHER NAME(S)(underline the oneyou prefer)

SCHOOL

YEAR OF STUDY S

FOR OFFICIAL USE

Marker

Marks1 2 3 4 5

TotalAGE

C U T A L O N G H E R EPlease write your solutions on A4 paper and staple the above form to them.

PLEASE WRITE YOUR NAME ON EVERY PAGE.Send your entry through your school to the section organiser.

For further information on the competition, please see the School Materials which have been distributed toschools. A copy of these Materials can be obtained from

http://www.wpr3.co.uk/MC/materials/index.htmlThere are separate links for primary and secondary schools. This page also includes a list of authorities ineach section and names and addresses of section organisers.

J4. At present, the sum of the ages of the parents, , is five times the sum of the ages of their children,. Two years ago, the sum of the ages of the husband and wife was eleven times the sum of the ages

of the same children. A year from now, it will be four times the sum of the ages of the samechildren.

PC

Determine the number of children.

J5. Two numbers and satisfy three of the following equations but do not satisfy the remaining one.x y

x + y = 63 x − y = 47 xy = 392x

y= 8

What is the value of ?x

END OF PROBLEM SET 1CLOSING DATE FOR RECEIPT OF SOLUTIONS :

5 November 2021

Look out for Problems 2 in late November!

For information about Mathematical Challenge, look on the SMC web site:

www.scot-maths.co.uk








Middle Division: Problems 1

M1. At present, the sum of the ages of the parents, , is five times the sum of the ages of their children,. Two years ago, the sum of the ages of the husband and wife was eleven times the sum of the ages

of the same children. A year from now, it will be four times the sum of the ages of the samechildren.

PC

Determine the number of children.

M2. Two numbers and satisfy three of the following equations but do not satisfy the remaining one.x y

x + y = 63 x − y = 47 xy = 392x

y= 8

What is the value of ?x

M3. Let the large square shown have side length 4. A second square is obtained by connecting the mid-points of the sides of the first square (as shown on the diagram).

If this process of forming smaller inner squares by connecting the mid-points of the previous squareis continued, what will be the side-length of the 12th square?

SEE OVER FOR QUESTIONS M4 & M5.

SMC

Mathematical Challenge Problems 1MIDDLE DIVISION 2021-2022


SURNAME


SCHOOL

YEAR OF STUDY S

FOR OFFICIAL USE

Marker

Marks1 2 3 4 5

TotalAGE





M4. A piece of A4 paper remains the same shape when it is folded in half i.e. the ratio of length tobreadth remains the same. Show that this ratio is .2 : 1Take a piece of A4 paper and fold as shown below to create a shape.

Prove that this shape is a kite and find its area.

M5. The sum of the cubes of three consecutive integers is 33 times the middle integer. Find the three integers.

END OF PROBLEM SET 1

CLOSING DATE FOR RECEIPT OF SOLUTIONS :

5 November 2021


For information about Mathematical Challenge, look on the SMC web site: www.scot-maths.co.uk








Senior Division: Problems 1

S1. A piece of A4 paper remains the same shape when it is folded in half i.e. the ratio of length tobreadth remains the same. Show that this ratio is .2 : 1Take a piece of A4 paper and fold as shown below to create a shape.

Prove that this shape is a kite and find its area.

S2. The sum of the cubes of three consecutive integers is 33 times the middle integer. Find the three integers.

S3. If and , and if and are real numbers, find the value of .f (x) = px + q f (f (f (x))) = 8x + 21 p q p + q

S4. 1 7 312 8 323 9 33

In each column of this grid the first number is the smallest number that cannot be achieved by addingtogether some combination of the numbers in the preceding columns. What will the numbers be inthe 4th column? What will the numbers be in the th column? Explain.n

SEE OVER FOR QUESTION S5.

SMC

Mathematical Challenge Problems 1SENIOR DIVISION 2021-2022


SURNAME


SCHOOL

YEAR OF STUDY S

FOR OFFICIAL USE

Marker

Marks1 2 3 4 5

TotalAGE





S5. An equilateral triangle has side length 2. A square, ,is such that lies on , lies on , and and lie on as shown. The points , , and move so that , and always remain on the sides of the triangle and moves from

to through the interior of the triangle. If the points , , and always form the vertices of a square, show that the path

traced out by is a straight line parallel to .

ABC PQRSP AB Q BC R S AC

P Q R S P Q RS

AC AB P QR S

S BC

A

B C

P

Q

R

S

END OF PROBLEM SET 1

CLOSING DATE FOR RECEIPT OF SOLUTIONS :

5 November 2021


For information about Mathematical Challenge, look on the SMC web site:


The Scottish Mathematical Council

MATHEMATICAL CHALLENGE (SECONDARY)

Registration Fee 2021−2022

Participating schools are required to pay a registration fee. The fee covers both sets ofproblems. The secondary school fee is £20 for 1-10 entrants in Round 1, £10 for the next 10 (orpart thereof) and so on. A cheque made payable to

SMC Mathematical Challengeshould be attached to this form (but see below) and

returned with the first set of problems.

School details Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Telephone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Address . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Education Authority (where appropriate) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contact teacher Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

email address . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(if used)

Payments: we prefer to be paid by cheque but if this is not possible we will accept payment byBACS. Our account is:SMC MATHEMATICAL CHALLENGEThe details are: 83-06-08 1054 8147.The reference you provide must include the school postcode followed by the school name.

Data ProtectionOur policy is attached. By completing and returning this form we will assume that you haveread it and accept it.

Submission of scriptsYou must submit on time and use correct postage. As a guide, first class large letters rates are:

Weight below 100g 250g 500g 750gPostage £1.29 £1.83 £2.39 £3.30

Data Protection and Privacy Policy for the SMC Maths Challenge

In the following, the data referredto is storedon the computerusedfor enteringpupil andschooldetailsandfor recordingpupil performance.Accessto all suchinformationis passwordprotected.

For each entrant the data stored is:

• surname,othername,school,year of study,and (after marking) the marksfor eachquestion. Permissionto record this data is given by completingthe tear-off formattached to the solutions.

For each school the data stored is:

• schoolname,address,phone,Primaryor Secondary,LEA (Local EducationAuthorityarea),whetherindependent,contactteachernameande-mail, andother informationyou want to add (if any).

Teachershaveaccess,by logging in with their usernameandpassword,only to the datafortheir own schoolandits entrants,andareresponsiblefor communicatingresultsandawardstotheir entrants.

SectionOrganisershaveaccess,by loggingin with their usernameandpassword,only to dataabout schools and entrants within their own Section.

Designatedmembersof the Maths ChallengeNational Committeehaveaccess,by logging inwith their usernameand password,to all the data in the databaseto facilitate running thechallenge.

No datawill be kept for more than 4 years.This would allow for productionof comparativereports over 3 years for the Scottish Mathematical Council Journal.

The data is stored on a secure server.

Contact: [email protected]

Last update: 7/8/21

Using the marks website

A Record of Entrants must be made on the marks website by the school or it will not be possible forthe section organiser to later add the marks for each entrant.

Go to the website: https://www.scottishmathschallenge.org.uk/

If you have used the website for your school in a previous year and canremember the username and password log in as usual.

Otherwise choose "Register here"

which enables you to enter your details and create a password. Yourusername will be your email address so please make sure that you enterthis correctly. The system will send you an email with an activation link:click on this link to verify your email address and then log in.

You will need your username and password each time you log in.

Once logged in click on the link

“Edit School Details"

Check that all the contact details are correct, making changes asnecessary. In particular, make sure that you select primary or secondaryand the relevant local authority whenever you make changes.

Later, when you have a pile of entries, or a list of entrants, log in againand go to "Add/Edit Entrants"and enter • the division (only for secondary pupils), • the names, and • the school year of each entrant.

When all your entrants have been added, to get a printout to send with your entries go to

"Printable version of details and entrants"

When the marks for each round are released, to view the marks for your school go to

"Marks page"

On the main page after you log in you may, from time to time, see messages from the sectionorganisers.

NOTE: If you experience difficulties with the website please contact either your section organiser or Helen Martin [email protected]



MATHEMATICAL CHALLENGE2021−2022

Forest Rovers, Southside United, Hilltown Thistle and ValleyWanderers were to play each other at football. After some of thematches had been played, a table showing some details ofmatches played, won, lost, drawn etc looked like this:

Played Won Lost Drawn Goals for

Goals against

Points

Forest Rovers 2 1 0

Southside United 0

Hilltown Thistle 1

Valley Wanderers 0 0 4 2 6

3 points are given for a win and 1 for a draw. Complete the table and findthe score in each match played, explaining how you worked it out.

See your teacher for details of the competition problems.



MATHEMATICAL CHALLENGE2020−2021

Did you solve it?

The owner of some stables has to fill in yet another form and sohe writes:

“My herd consists of horses and foals. A fifth of the herd is in the yard and a third of the herd is out topasture. These are all horses.Three times the difference are the foals, which are in the barn.The remaining two horses, in the paddock, belong to mydaughter.”

How many animals are in the herd altogether?

Solution

A fifth are in the yard and a third are in the pasture.

Difference is .1

3−

1

5=

2

15

Foals in barn are three times the difference which is .3 ×215

=615

=25

So in the pasture, yard and barn there are 1

5+

1

3+

2

5=

3

15+

5

15+

6

15=

14

15.

This means that the two horses in the paddock are of the

total.

1 −14

15=

1

15

So the total number of animals is 2 × 15 = 30.

See your teacher for details of the competition problems.

Mathematical Challenges VI 2003-2006 £8.00 ×

Mathematical Challenges V 2000-2003 £8.00 ×

Mathematical Challenges IV 1997-2000 £8.00 ×

Mathematical Challenges III 1994-1997 £7.50 ×

Primary Mathematical Challenges up to 2001 £5.00 ×

Total

Copies of any of these can be obtained from Bill Richardson, Kintail, Longmorn, Elgin IV30 8RJ

Cheques with order are preferred. Cheques should be made payable to 'SMC Mathematical Challenge'.

email enquiries:- [email protected] name and address:

Documents

Secondary Mathematical Challenges